Using integration, find the area of the region bounded by the lines 2y=5x+7, the x-axis and the lines 2y=5x+7, the x-axis, and the lines x=2 and x=8.

Given the boundaries of the area to be found are,

• The line equation is 2y = 5x + 7

• The y= 0, x-axis

• x = 2 (a line parallel toy-axis)

• x = 8 (a line parallel toy-axis)

As per the given boundaries,

• The line 2y = 5x + 7.

• x=2 is parallel toy-axis at 2 units away from the y-axis.

• x=8 is parallel toy-axis at 8 units away from the y-axis.

• y = 0, the x-axis.

• The four boundaries of the region to be found are,

Point A, where the line 2y = 5x + 7 and x=2 meet.

Point B, where the line 2y = 5x + 7and x=8 meet.

Point C, where the x-axis and x=8 meet i.e. C(8,0).

Point D, where the x-axis and x=2 meet i.e. D(2,0).

The line equation 2y = 5x + 7 can be written as,

Area of the required region = Area of ABCD.

[Using the formula and ]

The Area of the required region = 96 sq. units.

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